关于不定方程x2-7y4＝93

不定方程x2-Dy4＝C(其中D,C为给定的整数,且D＞0为非平方数)曾引起许多人的兴趣,Cohn,Tzanakis,黎进香等都对此类方程进行过研究.本文讨论了不定方程x2-7y4＝93正整数解的情况.所用方法是先用Pell方程将x2-7y4＝93的可能整数解进行分类,使其包含在几个式子里面,然后对这几个式子取模,借助于平方剩余的理论缩小解的范围,同时还利用了一些初等的证明方法,如递推序列,同余式.最后证明了不定方程x2-7y4＝93仅有正整数解(x,y)=(10,1),(130,7).

On the Diophantine Equation x2-7y4=93

The study of the diophantine equation x2-Dy4=C(D,C are given,and D is not a square integer) has caused some authors interest,such as,Li Jin-xiang and so on.In this paper,the author studies all the positive integer solutions of the diophantine equation x2-7y4=93.the process is as follows: classify the will-be integer solutions of the diophantine equation into four equations by Pell function firstly,then take models on these equations,so that the scale of the solutions will be reduced.At the same time,the methods of recursive sequence,congruence are used.At last,it is proved that the diophantine equation x2-7y4=93 has only positive integer solutions(x,y)=(10,1),(130,7).